THE ANALOGY OF A PLANE WORLD
At the risk of some prolixity I will go fully into the experience of
a hypothetical creature confined to motion on a plane surface. By so
doing I shall obtain an analogy which will serve in our subsequent
enquiries, because the change in our conception, which we make in
passing from the shapes and motions in two dimensions to those in
three, affords a pattern by which we can pass on still further to the
conception of an existence in four-dimensional space.
A piece of paper on a smooth table affords a ready image of a
two-dimensional existence. If we suppose the being represented by
the piece of paper to have no knowledge of the thickness by which
he projects above the surface of the table, it is obvious that he
can have no knowledge of objects of a similar description, except by
the contact with their edges. His body and the objects in his world
have a thickness of which however, he has no consciousness. Since
the direction stretching up from the table is unknown to him he will
think of the objects of his world as extending in two dimensions only.
Figures are to him completely bounded by their lines, just as solid
objects are to us by their surfaces. He cannot conceive of approaching
the centre of a circle, except by breaking through the circumference,
for the circumference encloses the centre in the directions in which
motion is possible to him. The plane surface over which he slips and
with which he is always in contact will be unknown to him; there are no
differences by which he can recognise its existence.
But for the purposes of our analogy this representation is deficient.
A being as thus described has nothing about him to push off from, the
surface over which he slips affords no means by which he can move in
one direction rather than another. Placed on a surface over which he
slips freely, he is in a condition analogous to that in which we should
be if we were suspended free in space. There is nothing which he can
push off from in any direction known to him.
Let us therefore modify our representation. Let us suppose a vertical
plane against which particles of thin matter slip, never leaving the
surface. Let these particles possess an attractive force and cohere
together into a disk; this disk will represent the globe of a plane
being. He must be conceived as existing on the rim.
[Illustration: Fig. 4.]
Let 1 represent this vertical disk of flat matter and 2 the plane being
on it, standing upon its rim as we stand on the surface of our earth.
The direction of the attractive force of his matter will give the
creature a knowledge of up and down, determining for him one direction
in his plane space. Also, since he can move along the surface of his
earth, he will have the sense of a direction parallel to its surface,
which we may call forwards and backwards.
He will have no sense of right and left—that is, of the direction which
we recognise as extending out from the plane to our right and left.
The distinction of right and left is the one that we must suppose to
be absent, in order to project ourselves into the condition of a plane
being.
Let the reader imagine himself, as he looks along the plane, fig. 4,
to become more and more identified with the thin body on it, till he
finally looks along parallel to the surface of the plane earth, and up
and down, losing the sense of the direction which stretches right and
left. This direction will be an unknown dimension to him.
Our space conceptions are so intimately connected with those which
we derive from the existence of gravitation that it is difficult to
realise the condition of a plane being, without picturing him as in
material surroundings with a definite direction of up and down. Hence
the necessity of our somewhat elaborate scheme of representation,
which, when its import has been grasped, can be dispensed with for the
simpler one of a thin object slipping over a smooth surface, which lies
in front of us.
It is obvious that we must suppose some means by which the plane being
is kept in contact with the surface on which he slips. The simplest
supposition to make is that there is a transverse gravity, which keeps
him to the plane. This gravity must be thought of as different to the
attraction exercised by his matter, and as unperceived by him.
At this stage of our enquiry I do not wish to enter into the question
of how a plane being could arrive at a knowledge of the third
dimension, but simply to investigate his plane consciousness.
It is obvious that the existence of a plane being must be very limited.
A straight line standing up from the surface of his earth affords a bar
to his progress. An object like a wheel which rotates round an axis
would be unknown to him, for there is no conceivable way in which he
can get to the centre without going through the circumference. He would
have spinning disks, but could not get to the centre of them. The plane
being can represent the motion from any one point of his space to any
other, by means of two straight lines drawn at right angles to each
other.
Let AX and AY be two such axes. He can accomplish the translation from
A to B by going along AX to C, and then from C along CB parallel to AY.
The same result can of course be obtained by moving to D along AY and
then parallel to AX from D to B, or of course by any diagonal movement
compounded by these axial movements.
[Illustration: Fig. 5.]
By means of movements parallel to these two axes he can proceed (except
for material obstacles) from any one point of his space to any other.
If now we suppose a third line drawn out from A at right angles to the
plane it is evident that no motion in either of the two dimensions he
knows will carry him in the least degree in the direction represented
by AZ.
[Illustration: Fig. 6.]
The lines AZ and AX determine a plane. If he could be taken off his
plane, and transferred to the plane AXZ, he would be in a world exactly
like his own. From every line in his world there goes off a space world
exactly like his own.
[Illustration: Fig. 7.]
From every point in his world a line can be drawn parallel to AZ in
the direction unknown to him. If we suppose the square in fig. 7 to be
a geometrical square from every point of it, inside as well as on the
contour, a straight line can be drawn parallel to AZ. The assemblage
of these lines constitute a solid figure, of which the square in the
plane is the base. If we consider the square to represent an object
in the plane being’s world then we must attribute to it a very small
thickness, for every real thing must possess all three dimensions.
This thickness he does not perceive, but thinks of this real object as
a geometrical square. He thinks of it as possessing area only, and no
degree of solidity. The edges which project from the plane to a very
small extent he thinks of as having merely length and no breadth—as
being, in fact, geometrical lines.
With the first step in the apprehension of a third dimension there
would come to a plane being the conviction that he had previously
formed a wrong conception of the nature of his material objects. He
had conceived them as geometrical figures of two dimensions only. If a
third dimension exists, such figures are incapable of real existence.
Thus he would admit that all his real objects had a certain, though
very small thickness in the unknown dimension, and that the conditions
of his existence demanded the supposition of an extended sheet of
matter, from contact with which in their motion his objects never
diverge.
Analogous conceptions must be formed by us on the supposition of a
four-dimensional existence. We must suppose a direction in which we can
never point extending from every point of our space. We must draw a
distinction between a geometrical cube and a cube of real matter. The
cube of real matter we must suppose to have an extension in an unknown
direction, real, but so small as to be imperceptible by us. From every
point of a cube, interior as well as exterior, we must imagine that it
is possible to draw a line in the unknown direction. The assemblage of
these lines would constitute a higher solid. The lines going off in
the unknown direction from the face of a cube would constitute a cube
starting from that face. Of this cube all that we should see in our
space would be the face.
Again, just as the plane being can represent any motion in his space by
two axes, so we can represent any motion in our three-dimensional space
by means of three axes. There is no point in our space to which we
cannot move by some combination of movements on the directions marked
out by these axes.
On the assumption of a fourth dimension we have to suppose a fourth
axis, which we will call AW. It must be supposed to be at right angles
to each and every one of the three axes AX, AY, AZ. Just as the two
axes, AX, AZ, determine a plane which is similar to the original plane
on which we supposed the plane being to exist, but which runs off from
it, and only meets it in a line; so in our space if we take any three
axes such as AX, AY, and AW, they determine a space like our space
world. This space runs off from our space, and if we were transferred
to it we should find ourselves in a space exactly similar to our own.
We must give up any attempt to picture this space in its relation
to ours, just as a plane being would have to give up any attempt to
picture a plane at right angles to his plane.
Such a space and ours run in different directions from the plane of AX
and AY. They meet in this plane but have nothing else in common, just
as the plane space of AX and AY and that of AX and AZ run in different
directions and have but the line AX in common.
Omitting all discussion of the manner on which a plane being might be
conceived to form a theory of a three-dimensional existence, let us
examine how, with the means at his disposal, he could represent the
properties of three-dimensional objects.
There are two ways in which the plane being can think of one of our
solid bodies. He can think of the cube, fig. 8, as composed of a number
of sections parallel to his plane, each lying in the third dimension
a little further off from his plane than the preceding one. These
sections he can represent as a series of plane figures lying in his
plane, but in so representing them he destroys the coherence of them
in the higher figure. The set of squares, A, B, C, D, represents the
section parallel to the plane of the cube shown in figure, but they are
not in their proper relative positions.
[Illustration: Fig. 8.]
The plane being can trace out a movement in the third dimension by
assuming discontinuous leaps from one section to another. Thus,
a motion along the edge of the cube from left to right would be
represented in the set of sections in the plane as the succession of
the corners of the sections A, B, C, D. A point moving from A through
BCD in our space must be represented in the plane as appearing in A,
then in B, and so on, without passing through the intervening plane
space.
In these sections the plane being leaves out, of course, the extension
in the third dimension; the distance between any two sections is not
represented. In order to realise this distance the conception of motion
can be employed.
[Illustration: Fig. 9.]
Let fig. 9 represent a cube passing transverse to the plane. It will
appear to the plane being as a square object, but the matter of which
this object is composed will be continually altering. One material
particle takes the place of another, but it does not come from anywhere
or go anywhere in the space which the plane being knows.
The analogous manner of representing a higher solid in our case, is to
conceive it as composed of a number of sections, each lying a little
further off in the unknown direction than the preceding.
[Illustration: Fig. 10.]
We can represent these sections as a number of solids. Thus the cubes
A, B, C, D, may be considered as the sections at different intervals in
the unknown dimension of a higher cube. Arranged thus their coherence
in the higher figure is destroyed, they are mere representations.
A motion in the fourth dimension from A through B, C, etc., would be
continuous, but we can only represent it as the occupation of the
positions A, B, C, etc., in succession. We can exhibit the results of
the motion at different stages, but no more.
In this representation we have left out the distance between one
section and another; we have considered the higher body merely as a
series of sections, and so left out its contents. The only way to
exhibit its contents is to call in the aid of the conception of motion.
[Illustration: Fig. 11.]
If a higher cube passes transverse to our space, it will appear as
a cube isolated in space, the part that has not come into our space
and the part that has passed through will not be visible. The gradual
passing through our space would appear as the change of the matter
of the cube before us. One material particle in it is succeeded by
another, neither coming nor going in any direction we can point to. In
this manner, by the duration of the figure, we can exhibit the higher
dimensionality of it; a cube of our matter, under the circumstances
supposed, namely, that it has a motion transverse to our space, would
instantly disappear. A higher cube would last till it had passed
transverse to our space by its whole distance of extension in the
fourth dimension.
As the plane being can think of the cube as consisting of sections,
each like a figure he knows, extending away from his plane, so we can
think of a higher solid as composed of sections, each like a solid
which we know, but extending away from our space.
Thus, taking a higher cube, we can look on it as starting from a cube
in our space and extending in the unknown dimension.
[Illustration: Fig. 12.]
Take the face A and conceive it to exist as simply a face, a square
with no thickness. From this face the cube in our space extends by the
occupation of space which we can see.
But from this face there extends equally a cube in the unknown
dimension. We can think of the higher cube, then, by taking the set
of sections A, B, C, D, etc., and considering that from each of them
there runs a cube. These cubes have nothing in common with each other,
and of each of them in its actual position all that we can have in our
space is an isolated square. It is obvious that we can take our series
of sections in any manner we please. We can take them parallel, for
instance, to any one of the three isolated faces shown in the figure.
Corresponding to the three series of sections at right angles to each
other, which we can make of the cube in space, we must conceive of the
higher cube, as composed of cubes starting from squares parallel to the
faces of the cube, and of these cubes all that exist in our space are
the isolated squares from which they start.